An exponential approach to highly ill-conditioned linear systems

We propose and analyse an exponential approach to solving highly ill-conditioned linear systems with a positive definite matrix. This approach also provides a new technique to find the inverse of a given positive definite matrix by a one-stop procedure. The advantages of this approach are its conceptual simplicity and ease of implementation. Numerical experiments are carried out. Numerical results demonstrate the effectiveness and superiority of the exponential approach to solving highly ill-conditioned systems of algebraic equations in comparison with the standard MATLAB codes.